The result of Theorem REF  severely constrains the spacetimes with locally rotational symmetry that can admit dynamical horizons and timelike membranes. In particular, Theorem REF  tells us that, for spacetimes with locally rotational symmetry where all vector and tensor quantities vanish identically, rotation and spatial twist are obstacles to the existence of truly dynamical black holes. Hence, only the class II of these spacetimes with \(\Omega =0\)  and \(\xi =0\)  admit such horizons. Well known examples of this class include the Lemaitre-Tolman-Bondi (LTB) solution, Schwarzschild spacetime (although the MTT here, a null horizon, is minimal since both expansion scalars are identically zero), the Oppenheimer-Snyder (OS) dust solution and the Robertson-Walker (RW) solution. (The LTB and RW solutions, for example, contain dynamical horizons foliated by non-minimal MOTS, and the OS dust collapse contains timelike membranes, also foliated by nonminimal MOTS. (These were recently studied in context of the 1+1+2 formulation [1]}.)
